MINISTRY OF SUPPLY

AERONAUTICAL RESEARCH COUNCIL

' REPORTS AND MEMORANDA

The Simple Harmonic Motion ofa Helicopter

Rotor with Hinged Blades By

J. K. ZBROZEK, Dipl.Eng.

Crown C@yright Reserved

LONDON" HER MAJESTY'S STATIONERY OFFICE

1955

P R I C E 3S 6d N E T

R. & M. No. 2813 (12,466)

A.R.C. Technical Report

The Simple Harmonic Rotor with

Motion Hinged By

J. K. ZBxozE~:, I)ipl.Eng.

COMMUNICATED BY THE PRINCIPAL DIRECTOR OF SCIENTIFIC RESEARCH (AIR), ~/~INISTRY OF SUPPLY

of a Helicopter Blades

Reports and Memoranda No. 2 813 *

April, 1949

Summary.--In simple harmonic oscillation of the helicopter with hinged blades, the tip-path plane is tilted with respect to the shaft in the plane of oscillation and in the plane perpendicular to it. The angles of tilt can be ex- pressed as functions of angular velocity and acceleration. The influence' of the acceleration term on the dynamic stability of the helicopter is small.

The expressions for angles of tilt due to angular velocity can be simplified to the expressions obtained in previous work under assumptions of quasi-static conditions.

1. I~troductio~.--It has been shown in Ref. 1 that when the rotor shaft of a helicopter tilts with constant pitching velocity, the t ip-path plane lags behind the shaft and also tilts Sideways. These angles of lag and sidetilt were found to be proportional to the pitching velocity of the shaft and the following expressions were derived:

Oal _ _ 1 6 . Of yD . . . . . . . . . . . . . (1)

~b I 1 . ~q -- 9 , . . . . . . . . . . . . . . (2)

where al is the angle of lag between the rotor shaft and the t ip-path plane axis measured in the pitching plane and q is the pitching velocity ; bl is the angle of sidetilt, y is Lock's inertia number and ~ is the rotational velocity of the rotor.

In I~. & M. 25091 the value of aa~/aq as given by equation (1) was used in the analysis of dynamic stability. Strictly speaking equation (1) applies only to a motion with constant rate of pitch. Where the motion of the helicopter approximates to a simple harmonic motion the use of the derivative given by equation (1) may not be justified in the study of the dynamic stability.

The present note gives a more detailed analysis of rotor derivatives and shows tha t the expressions (1) and (2) for rotary derivatives are accurate enough for any practical application. However, the analysis indicates the existence of acceleration derivatives, but their influence on the dynamic stability of the helicopter is small.

* R.A.E. Report Aero. 2319, received 14th July, 1949.

1

2. The Approach to the Pfoblem.--It is assumed tha t the helicopter is hovering with its shaft vertical and t ip-path plane horizontal. Suddenly the helicopter and its shaft begin to oscillate in a pi tching plane about the rotor centre according to the equation:

0 = A s i n ~ t ; . . . . . . . . . . . . (3)

where A is some arbitrary amplitude and v the circular frequency of rotor shaf t oscillation.

The main difficulty in the present problem is estimating the magnitude and direction of the flow through the disc. Lacking any data it is assumed tha t the flow through the disc is in the direction opposite to the instantaneous direction of lift. In other words no allowance for any downwash lag is made.

I t is further assumed tha t the blade hing e has zero offset, and the hinge bearing has no friction.

The diagram of axes and angles is shown in Fig. 1. The reference axes are chosen as horizontal and vertical, i.e., the axis of the rotor in its initial conditions. All the values are measured from these axes, positive in an anti-clockwise direction for Fig. 1. The following quantities are defined:

0 the angle of the shaft tilt measured from the vertical and given by equation (3)

/3 the flapping angle measured between blade axis and horizon

/la the angle of lag, between the shaft and the t ip-path plane axis, positive in positive direction of 0.

The equation of motion is written, assuming tha t all the quantities are measured from OX-plane.

Due to the blade hinge being fixed relative to the shaft, the tilt of the shaft in the pitching plane by an angle 0 produces, in the horizontal plane, feathering in the form:

-- 0 sin W . . . . . . . . . . . . (4)

where w = fat is the blade azimuth angle measured in the 0X-plane from the downwind blade position. Tile angle of shaft tilt 0 is a function of time given by equation (3) and hence the feathering is given by:

- - A sin vt sin fat . . . . . . . . . . (5)

and the blade pitch angle by:

@' = @0 -- A sin vt sin fat . . . . . . . . . . (6)

where ~'0 is the collective pitch angle.

The moments acting on the blade about the flapping hinge are as follows:

Moment due to centrifugM force:

- - 1 # 2 / ~ . . . . . . . . . . . . . (7)

Moment due to angular acceleration of blade in flapping motion:

- z,p . . . . . . . . . . . . . . . ( s ) 2

The aerodynamic moment :

I~½pa@o-- A s i n v t s i n ~ t .Qr

where U is veloci ty of flow th rough and perpendicular to the disc. I t is assumed tha t the coning angle a0 is small and tha t cos a0 can be taken as unity. The flow velocity U is t aken to be constant . In tegra t ing the aerodynamic m o m e n t along the blade and in t roducing Lock's iner t ia number , the aerodynamic m o m e n t can be expressed as:

where

D A sin ~,t cos t?t) . . . . . . . . . . (10)

PagR4 . . . . . . (11) Y -- I i . . . . . .

is Lock's inert ia number• g is a mean chord the exact value of which can be defined by compari- son of equat ions (9) and (10), but with sufficient accuracy the mean chord can be defined as '

f~r:~dr (12)

is the coefficient of the flow through ~the disc defined as:

= u / ~ R . . . . . . . . . . . (13)

In the evaluat ion of integral (9) no allowance has been made for t ip and root losses.

N e g l e c t i n g the gravi ty term, the equat ion of mot ion of the blade is obta ined from the ex- pressions (7), (8) and (10)

1 2 fi + ~),t?~ + ~92~ ~),9 (~6 -- ~Z) - - ~,~9eA sin vt sin t?t . . . . . . . (14)

Remember ing tha t the coning angle a0 is given by R. & M. 1127 ~ as

ao = ~7(Oo - ~ z ) ; . . . . . . . . . . (15)

and int roducing t r igonometr ical subst i tut ions equat ion (14) can be put into the following al ternat ive form for ease of in tegrat ion

= ~ A-f2 2 (~? v)t ¢~A7£2 ~ (f2 - - v)t. (16) fi + }~X?t) + Y2~ a0 ~2 + ~ r cos + - - cos . .

The complete solution of equat ion (16) takes the form:

/3 = ao + e~{C~ cos vFt + C2 sin ~ t }

+ C8 cos (t9 + ~)t + C4 sin (sO + ~,)t

+ "C5 cos ( n - - ~)t + C6 s i n (n - ~)t ; . . . . . . . .

where C1, C.,, etc., are constants of integrat ion and are functions of A, ~?, ~ and ~, only.

3

(17>

The first part of equation (17) shows that due to an initial disturbance the blade will osciliate with frequency given by

. . - ( i s )

This oscillation is very heavily damped and the damping factor is given by:

2~ - - yY2 16 . . . . . . . . . . . . (19)

After the blade motion due to the initial disturbance is damped out, which in practice takes about ~ sec, the mean value of blade flapping angle approaches ao, the coning angle. The second part of equation (17) shows tha t the blade will oscillate steadily about its flapping hinge with two frequencies, ~? + v and ~ -- v.

After some re-arrangement of equation (17) and omitting the rapidly damped part of the motion, the expression for steady blade flapping measured from horizontal will be in the form:

fi' = a0 + (C~ cos vt + Ca sin vt) cos ~ + (C9 cos ,t + C10 sin vt) sin ~ ; . . . . (20)

where ~0 = ~?t and C~, Ca, etc., are constants and functions of A, ~9, , and 7.

The coefficient of cos ? gives the longitudinal ti l t of the t ip-path plane and the coefficient of sin ~0 the lateral t i l t of the t ip-path plane, both measured from horizontal.

The angle between the shaft and 0Z-axis as given by equation (3) is

0 = A s i n v t . . . . . . . . . . . . (3)

and the angle of the longitudinal la G i .e . , the angle between the t ip-path plane axis and shaft axis (positive in positive direction of 0) i s : - -

Remembering that :

A a = (C7 cos vt + Ca sin vt) - - A sin yr.

0 = A sin vt ]

O = A v cos vt

0 = - - A v 2sinvt

(21)

(22)

and putt ing

v / t ) = ~ . . . . . . . . . . . . (23)

the angle between the shaft and t ip-path plane axis, measured in the plane of the shaft oscillation is given by the expression:

A a - -

1 1 (24)

4

The corresponding lateral t i l t of t ip-pa th plane is given by:

1 ( y y { ( , ) ~ _,)1

1 (~,'~((r'@_~.,.) + ~(4 _~,)11 o. o ~ \ g / i \ g / ; ~ ' " . . . . .

and the tilt of the rotor t ip-path plane is positive towards the advancing blade. (24) and (25), D is an abbreviat ion for the following expression:

- D = (1 V')" + 2V' (4 - - a r e + 5~) + ~(4 - - . . .

. . (2s)

In equat ions

.. (26)

3. Discussion.--3.1. Longitudinal Disc Tilt. The equat ion (24) can be rewri t ten:

16AO + 1 f(16"~ 2 ~ a - g o ~ i \ 7 / - 1 } t o ; . . . . . . . . (27)

where A a n d / " are functions of ~, and ~ only, and for small values of 5 = v/fa can be taken as unity. The full expressions for A a n d / 1 are a s fol lows:--

, j - ~ , • . . . . . . .

@ ' ( ! (~)~(1 - - V') - - (~ )"(4 - - 3V' + 25 ~) - - 52(4 - - 5~) 2} 8 / (

; . . . . (29) f / r V - - 4~ X D J

where D is given by the equat ion (26).

The numerical values of these functions are given in Fig. 2, and were calculated for a range of values of ~ and y. The values of ~ as found in practice are usually very small ; for example in the case of the Sikorsky R4-B helicopter, which has ~ = 13, the values of ~ are about 0 .02 for longitudinal and about 0 .05 for lateral oscillations. The value of ~ is seldom expected to exceed 0.05 for full-scale rotors. Also, for most of the small helicopter models the value of

= ~/fa remains of the same order due to the large values of ga necessary if the corresponding t ip speed is, to be main ta ined on the model: I t can be seen from Fig. 2 tha t wi th in the practical range of values the ratio of oscillations, ~ has only a small effect upon the values of rotor deriva- tives. Only for .very heavy blades, y < 6, can the Values of A a n d / ~ differ appreciably f rom unity, and in those cases for the purpose of s tabi l i ty calculations the corrections of Fig. 2 can be used.

For normal values of blade inert ia number the following approximat ion is justified:

A = r = 1 . . . . . . . . . . . . . (3o)

and thus simplify the equat ion (27) to the form:

Aa-, , -_160 I + - 1 } o . . . . . . . . ( a ) y~ £22 ( \ y .] •

5

The derivative of the tilt of the t ip-path plane relative to the shaft with respect to angular velocity of pitch is:

aal _ 16 . . . . . . . . . . (32) ~q yf2 '

the t ip-path plane lagging behind the shaft. The equation (32) is in agreement with equation (1), which was obtained from the simplified analysis of Ref. 1.

I

The derivative of the til t of the t ip-path plane relative to the shaft with respect to angular acceleration of pitch is:

a< l I(lS ' ! 1 - > L \ 7 / - t ) = v T/ (33)

The approximate expression for the acceleration derivative (33) and the expression for the correction factor F (29) are valid only for 7 =# 16 ; for the values of ~ equal or near 16 the full expression (24) for the acceleration derivative has to be used.

I t is interesting to note tha t the angular velocity derivative, equation (32), is always negative and always produces a damping of the angular motion of the helicopter. The sign of the angular acceleration derivative however depends on the value of blade inertia number. For practical values of blade inertia number (~ < 16) the acceleration derivative is positive, as shown by equation (33), i . e . , the apparent displacement derivative is negative, and it can be shown tha t the positive acceleration term has a stabilising effect on the helicopter stability. However the numerical calculations based on the Sikorsky R4-B helicopter show a very small effect due to this acceleration derivative. I t has to be borne in mind tha t the present analysis does not take into account any effect of the time lag of the aerodynamic forces acting on the blade. This effect could be of considerable importance in determining the blade motion.

For very light blades, y > 16, the acceleration derivative changes sign and becomes negative*. I t is worth noting tha t according to equation (18) for values of inertia number y > 16 the disturbed flapping motion of blade ceases to be oscillatory and becomes a pure subsidence.

To give a better illustration of the disc and shaft behaviour in simple harmonic motion, the diagrams of Fig. 3 were prepared. The thick line represents motion of the rotor shaft according to the equation"

0 = A s i n ~ , t ; . : • . . . . . . . . . . . (3)

The dotted line shows motion of the t ip-path plane with respect to the horizon. The angle c~ is the angle between t ip-path plane axis and the vertical. The motion of the t ip-path plane can be described by all equation" "

= CA sin (vt + 6) ; ' • . . . . . . . . .

The constants # and ¢ can be found from equation (20) and are functions of ; and y only.

(34)

For the practical range of values of y and 7, the motion of the rotor disc is shown in Fig. 3, top diagram. The ratio of amplitudes, # is slightly less t h a n unity, but for practical purposes can be taken as unity. The phase angle ¢ is small and negative. For infinitely heavy blades,

* The exac t va lue of y a t which the acce le ra t ion de r iva t ive becomes nega t ive is s l igh t ly less t h a n 16 and depends on the va lue of ~.

6

--> 0, Fig. 3, bottom diagram, the amplitude ratio # approaches zero value, and at the same time the phase angle ¢ ---> 90 deg. For the limiting case when ~ = 0 and # = 0, the t ip-path plane remains horizontal, irrespective of the shaft motion, which is quite obvious for physical reasons. For very light blades, when the inertia number ~ is large, the amplitude ratio ~ is increasing and becomes larger than uni ty (Fig. 3, bottom diagram). Increasing the value of ~ to infinity brings the value of # back to unity. The phase angle ¢ is decreasing steadily with increasing ~,, and for ), --> ~ the phase angle approaches zero. For infinitely light blades, # = 1 and ¢ = 0, the rotor disc follows exactly the motion of the shaft.

I t might be of interest to point out that the vertical shift between the thick and dotted lines of Fig. 3 is proportional to the tilt of the disc with respect to the shaft, and at the point vt = 0 this sh i f t i s proportional to the velocity derivative and at the point vt = ~/2 to the acceleration derivative. Bearing this in mind, it can be deduced immediately from Fig. 3, that the angular velocity derivative always remains negative, but the acceleration derivative may change s ign depending on the value of the inertia number.

To give some illustrations of the values involved, Fig. 4 was prepared, where the value of the amplitudes ratio, ~, is plotted against inertia number, ~, for different values of frequency ratio, 5. The scale to the right is too small ±o show the shape of the ~-curve for verY large values of 9,, and the dotted line shows only the tendency of the ~-curve.

Remembering that the values of ~ and r likely to meet in practice are 5 < 0.05 and ~ > 5, it can be seen tha t tile value of ¢ differs from uni ty by less than 1 per cent.

3.2. Lateral T i l t . - -For the practical range of values of ~ = ~/£2 the equation (25) for the lateral tilt of the t ip-path plane can be simplified to:

= Lo 1 240 = 1_6 + _3_1 0 . . . . . . . . . (35) g? X? :~ y D 2Q ~q

During the longitudinal oscillations of the helicopter with frequency v, the t ip-path plane tilts not only in the plane of oscillations, but in the plane perpendicular to it as well. This angle of tilt, A b, can be split up into two parts as in equation (35), first that due to angular velocity and second that due to the angular acceleration of the helicopter.

The corresponding derivatives of lateral tilt can be written:

Ob~ 1 ~q ~9

(36)

The value of 3bl/aq for constant rate of helicopter pitch was deduced in Ref. 1, Appendix II, from kinematic considerations, and is in agreement with present result.

Similarly the acceleration derivative of the sidetilt is:

~ b ~ = _ 1 2 4 _ 3 l Oal . . . . (37) ~q ~2 9, 2 t? Oq . . . .

4. Co~clusio~s.--The mathematical analysis of the dynamics of the flapping blade during -simple harmonic motion of the helicopter shows the existence of angular velocity and angular acceleration derivatives. The effect of the accleration derivative on the dynamic stabili ty of the helicopter is quite small, and for the usual values of blade inertia number it can be neglected. The effect of the frequency ratio 7 = v/X? on the values of the rotor derivatives is also small, mainly due to the fact tha t for all helicopters and for helicopter models b~tilt to dynamic a~d aerody~¢amic scale, t he value of ~ is likely to be much less than 0.1.

It seems that only for the case of helicopters with very heavy blades, say ~ < 8, need the effects of the acceleration term and of !requency ratio be taken into account.

I t has to be borne in mind that the analysis given here does not take into account any changes in slipstream structure due to the oscillation of the helicopter ; and further, the analysis covers only simple harmonic motion in hovering.

I t is proposed to analyse other modes of helicopter motion in hovering, say an exponential mode of motion, corresponding to motion due to stick displacement.

A further step will be the extension of the present analysis to forward flight.

No. Author

1 J . K . Zbrozek ..

2 C . N . H . Lock ..

3 J. K. Zbrozek

4 Miehl ..

• °

REFERENCES

Title, etc.

.. Investigation of Lateral and Directional Behaviour of Single Rotor Helicopter (Hoverfly Mk. I). R. & M. 2509. June, 1948.

• . F u r t h e r Development of Autogiro Theory. R. & M. 1127. March, 1927.

.. Control Response of Single Rotor Helicopter. A.R.C. 12,465 (To be published). March, 1949.

.. The Aerodynamics of the Lifting Airscrew with Hinged Blades in Curvi- linear Motion. Trans. Inst. Aero-Hydro@namics, No. 465, pp. 1-60. R.T.P. Translation No. 1533/M.A.P. 1940.

8

a

ao

a~

b~

B

c

I1

q = O

t

R

U

pagR ~

0

~9 0 .

U X2R

~F

VF

9~

P

= X2t

X2

A

T qb

$

List of Symbols

Lift slope of blade section

Coning angle

Coefficient of Fourier's series for flapping angle, and corresponding to backward tilt of the rotor disc

Coefficient of Fourier's series for flapping angle, and corresponding to lateral tilt of the rotor disc, positive towards tile advancing blade.

Tip-loss coefficient ; usually defined: B = 1 -- c/2R

Blade chord

Mean blade chord

Blade moment of inertia about flapping hinge

Pitching velocity of helicopter

Time

Rotor radius

Velocity of flow through the disc

Angle between rotor disc and horizon

Flapping angle, measured between longitudinal axis of the blade and horizon

Blade inertia number

Angle of shaft tilt in pitching plane and measured from vertical

Pitch angle of rotor blade ; measured from blade airfoil chord

Collective pitch

Coefficient of flow through the disc

Damping coefficient of disturbed flapping motion

Frequency of disturbed flapping motion

Circular frequency of shaft oscillation

Frequency ratio

Air density

Blade azimuth angle, measured from rearmost position

Rotational velocity of rotor

Correction to angular velocity derivative

Correction to angular acceleration derivative

Ratio of disc to shaft amplitudes

Phase angle between shaft and disc motions

9

A P P E N D I X

The Compariso~ with Calculations for Co~stant A~gular

Velocity a~d Co~sta~t A~gular A cceleratio~,

Since the completion of the present report, the attention of the author was drawn to the work of Miehl, Ref. 4. Miehl extended Lock's analysis (R. & M. 11272) to steady curvilinear flight, and his investigations lead to the following expressions for the rotor tilt derivatives:

~al _ 16 1 ? B (B 2 - - 1/S

~ b l 1 1 / ~q fa 1 - - ff~/2B ~ J

(43)

where B is the usual tip-loss coefficient. The formulae (43) are in agreement with the present report if we assume no tip loss, B = 1, and very slow oscillation, ~ =- 0. " It can be seen from equation (43) that the forward speed has a very small effect on the values of the angular velocity derivatives.

Miehl (Ref. 4) in his work is considering one particular case of blade motion during uniformly accelerated rotation of helicopter in hovering. The formulae for the rotor tilt derivatives due to constant acceleration are in agreement with the present report, assuming frequency ratio, ~=-0 .

The comparison of the theoretical calculations with experimental values obtained from model tests, Ref. 4, have shown that in order to obtain satisfactory agreement it was necessary to modify the distribution of the induced velocities. It was assumed in Ref. 4 that the increments of induced velocity are proportional to the new mass forces present in curvilinear accelerated flight.

The calculations have shown that the re-distribution of induced velocities has a small effect on the values of angular derivatives, and is increasing slightly the value of ~al/~q, the value of ObjOq not being affected. However, re-distribution of induced velocities has a considerable effect on the acceleration derivative, and increases the value of ~al/O(t; the value of Obj~O is not affected. The assumed distribution of induced velocities gave good agreement between theory and experiment.

10

FIG. I .

z

l'ZoN

Diagram of rotor in simple harmonic motion.

I , I

A

l 'O

0-9

O-B

0.7

I . I

/-,

I ' 0

j l

o . I

~ - ¥ = I 0

0"2 :~

0.9

0,1

0."

]

. . . . i o o.t 0.2 ;~

- ~= I0

-~--8

"-~'=6

FIG. 2. Corrections to velocity and acceleration derivatives.

11

g 0,:. ,SHAFT Dt,,SPLAEEh-IENTt 0

/ PRACTICAL _VALuEs ~ " \

,SHAFT DISPLAcE~E.NT. e t" ~ ~ / 'T IP PATH PLANE

/ " ~ \ \ .A"-~TIP PATH PLANE.

/ / / " DISPLACEMENTp o~ ~'~ .

Y Attitude v e r s u s time c u r v e s of rotor

shaft and tip-path plane axis in steady harmonic motion.

FIG. 3.

H ̧ ¢~

0.9

o * 8

FIG. 4.

7

/ "~ = 0, I

9=0 A~D 0.01

5HAFT e = A S i n " ~

"rip PATH PLANE o( = ~A Sin ('0L + ¢)

IO 15 20 ~ "

Ratio of tip-path plane to shaft amplitudes in simple harmonic oscillation.

12 (5115) Wt. 17/680 K.7 10/Ss H.P.Co. S4-261 PRINTED IN GREAT BRITAIN

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